Washington (CNN)House and Senate Republicans are quickly being selected as delegates to the Republican National Convention this summer, potentially giving them tremendous sway in helping choose the party's presidential candidate if Donald Trump fails to clinch the nomination outright.

And that could be a challenge for Sen. Ted Cruz, who has labeled many of his Capitol Hill colleagues as part of a corrupt "Washington cartel."

In Kentucky, Senate Majority Leader Mitch McConnell was selected over the weekend as a delegate to the convention in Cleveland this summer. Trump supporter Sen. Jeff Sessions of Alabama is a delegate as well. And in North Carolina, a member of the House GOP leadership, Rep. Patrick McHenry, is expected to be a delegate, as is Sen. Thom Tillis.

In interviews with CNN, a wide range of lawmakers refused to say how they'd vote once most delegates are free to vote their conscience after a first round of balloting.

"Depending on how some decisions are going to be made, I could be part of helping the delegates as a whole make that decision" of selecting the nominee, Tillis said. "I'm legitimately uncommitted."

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The GOP lawmakers' presence adds a new wrinkle to the prospects of an open convention in July. While many congressional Republicans are planning on skipping what could be an unruly convention to worry about their own reelection campaigns, others are poised to play an outsized role if no candidate can secure the 1,237 delegates to clinch the nomination before July.

While Cruz has succeeded by courting an array of delegates who back his candidacy, many are simply undecided about how they'd vote in a contested convention -- including influential members of Congress.

McConnell, for one, could hold sway with many of his state's 46 delegates. The GOP leader, who has been a frequent Cruz punching bag, has been stressing electability over ideology -- that the party should nominate a candidate who could defeat the Democratic nominee, likely Hillary Clinton. That mirrors an argument that Ohio Gov. John Kasich, in particular, has been making as he tries to appeal to delegates.

"It could be a really interesting convention," McConnell told Kentucky Republicans attending their state party's convention in Lexington on Saturday. "One thing I want to assure you is there isn't any sort of insider effort that could go on to keep this from being anything other than a pretty open, transparent process."

The delegation includes Gov. Matt Bevin, Sen. Rand Paul and House members, including Rep. Andy Barr, an ally of McConnell's.

"He's uncommitted," Barr spokesman Rick VanMeter said when asked how his boss would vote after a first ballot.

What could give lawmakers some cover from angry constituents is the likelihood that their individual votes could be shielded from public scrutiny. While each state will publicly announce how many votes it will cast for each candidate, it may not be clear how individual delegates vote -- and that includes many lawmakers.

"I'm always amused by the suggestions that there is some kind of group that can deliver the nomination," McConnell said. "Let me say, if there were such a group, I probably would be in it, right? There isn't any such group."

What happens to Rubio delegates?

A big question remains what will happen to the 173 pledged delegates held by former GOP presidential candidate Marco Rubio once they become unbound. Cruz has claimed a number of them will back him on a second ballot and beyond, but Rubio could have a huge influence if he urges his delegates to vote a certain way.

In a brief interview last week, the Florida senator told CNN: "I haven't had those conversations" about whom his delegates should back. Rubio has not endorsed any of the candidates since dropping out of the race last month.

Asked if he'd seek his party's nomination if the convention remained deadlocked, Rubio was coy.

"I've spent no time thinking about that; I'm now focused on the Senate," he said.

States vary in how they handle candidates no longer in the race. In Oklahoma, delegates are bound by the results of the state, which Cruz carried with 34 percent of the vote, giving a boost to the Texas Republican and potentially Trump, who claimed second place with 28 percent. Yet party rules free Oklahoma delegates once a candidate's name no longer appears on the ballot for nomination at the convention -- something that could impact delegates for Rubio, who won 12 delegates by his third place showing in the state.

Each state has its own rules for when delegates are allowed vote their conscience. In the first round of voting, 95 percent of delegates are bound by the results of their states, a number that drops to 40 percent in the second round and 20 percent in a third round. In some states, rules make it harder for delegates to vote freely after the first ballot. In Alabama, where Trump won 36 delegates with 43 percent of the vote, the state's delegation is bound by the results until two-thirds of the delegates agree to release themselves and vote how they want.

Sessions, the Alabama Republican and Senate's lone GOP Trump supporter, said he doesn't expect mass defections from his delegation on the convention floor.

"We met with our delegates," Sessions said, himself a delegate. "I don't expect any of the Trump delegates in Alabama to alter their support for Trump."

Then there are the wild cards.

Oklahoma Sen. Jim Inhofe, a former Rubio supporter who now backs Kasich, has been sharply critical in the past of Cruz's sharply worded comments about fellow Senate Republicans. While he's not a delegate, Inhofe said he may urge a Trump-Kasich ticket in Cleveland.

"He'd never say that he'd be agreeable to be vice president but that's because that time hasn't come yet," Inhofe said of Kasich.